DIAGNOSIS Coil whine from the internal power supply transformer or a capacitor. It's drawing power constantly via the C-wire.
FIX This is often a defect in that specific unit's components. First, ensure the wires at the wall plate are tight. A loose wire can cause arcing. If tight, the noise is internal.
IF THAT DIDN'T WORK The only fix is unit replacement. The buzzing is a physical vibration of components. Contact Emerson support. They are aware of this issue in some early production runs.
HEADS UP Do not open the thermostat case. You will void the warranty and expose yourself to live electrical terminals.